3 mistakes learners make with vector paths, each one named and explained.
Travel back along an arrow and its vector changes sign.
b minus a is offered where the vector to a point inside or along the diagonal was wanted.
b minus a is the expression this topic produces most often, so it becomes the default answer.
Question. OM, where M is the midpoint of AB
A common answer. b - a
The answer. (1/2)(a + b)
Why. b - a is the vector AB itself, which runs along the side, not from O.
Say where the vector starts and where it ends. That decides which letters appear.
AB is given as a + b, with the direction along OA never reversed.
Adding along a path is the right idea, and only the direction of the first leg is wrong.
Question. AB, where OA = a and OB = b
A common answer. a + b
The answer. b - a
Why. Going from A to B means travelling backwards along a first, which makes it minus a.
Trace the journey with a finger. Every leg you travel backwards gets a minus sign.
The midpoint of AB is given as a + b rather than half of it.
Getting to a + b is the substantial step and it looks like a finished expression.
Question. M is the midpoint of AB, OA = a, OB = b
A common answer. a + b
The answer. (1/2)(a + b)
Why. a + b would take you past B altogether, not to the middle.
A midpoint is always an average, and an average always divides.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.